arXiv:math-ph/0310010v1 8 Oct 2003 where rms w frames two frames, e classes e U space ξ E H o oooia space topological a For Orientation 1 HOY okhpi ooro rf .A eAcraa ue9- June Azcarraga. de A. J. Prof. (Spain) of honour in Workshop THEORY. ′ ′ ( ∗ ∈ = in ⊂ ∗ ,F M, eatmnod ´sc eoia nvria eZaragoza F´ısica de Te´orica, de Universidad Departamento ( Let ob ulse ntePoednso:SMERE N RVT NF IN GRAVITY AND SYMMETRIES of: Proceedings the in published be To ,A X, btutos xesosadReductions. and Extensions Obstructions, H g ˆ M U oooyt hsclpolm,i seilt eue holon reduced to especial in applicat problems, M some physical explain to we etc., homology structures spin and entation M e · 1 c ′ ( i.e. , e M sa extension an is ) and - en nqeelement unique a define ( ,wt coefficients with ), ,G M, noted , as , = fe nrdcn oechmlg lse sobstructions as classes cohomology some introducing After ξ oeapiain fCohomology of applications Some ) g eamnfl fdimension of manifold a be ∈ · GL n F .Teipratojcshr the here objects important The ). e H -theories. nteoelpo w ace.Amnfl is manifold A patches. two of overlap the in needn etrfilsa n on in point any at fields vector independent ξ ∗ ( csfel in freely acts : ,A M, F e (in → -00 aaoa Spain Zaragoza, E-50009 ). X U E E and ) h motn bet r the are objects important the , → coe 2 2018 22, October ihfiber with [email protected] A M usJ Boya J. Luis { eeal in generally e e ′ g } Abstract n ti tefa itself is it and (in An . ftegnrllna group linear general the of 1 U F ′ orientation en ntesm ls fdet if class same the in being ) n atduo yagroup a by upon (acted osdraframe a Consider . Z h nees A integers. the , ehchmlg element, cohomology Cech ˘ characteristic in M sagoa ls of class global a is U 1 03 Salamanca, 2003. 11, oooygroups homology w frames Two . orientable oso co- of ions e GL m in omy cohomology oori- to G napatch a in bundle ∗ ( vran over ) ,R n, fi is it if > g IELD by ) . ξ e 0 , , : possible to give (globally) an orientation; chosing an orientation the manifold becomes oriented. So the questions are: first, when a manifold is orientable and second, if so, how many orientations are there. These questions are tailor-made for a cohomological answer. This is about the easiest example traslatable in simple cohomological terms by obstruction theory. Let τ be the principal bundle of the tangent bundle to M, so the total space B is the set of all frames over all points:
τ : GL(n, R) → B → M (1)
Matrices in GL+ with det > 0 have index two in GL, hence are invariant, with Z2 as quotient: we form therefore an associated bundle w1 = w1(τ):
GL+ ↓ τ : GL → B → M (2) ↓ ↓ k w1 : Z2 → B/2 → M Our space M is orientable if the structure group reduces to GL+. The set of principal G-bundles over M is noted Hˆ 1(M,G) and it is a cohomology set (in Cech˘ cohomology) [1] . Thus τ ∈ Hˆ 1(M,GL), and we have associated 1 to τ another bundle, name it Det τ ≡ w1(τ) ∈ H (M,Z2), called the first Stiefel-Whitney class of τ (as a real vector bundle). The Cech˘ cohomology set Hˆ 1 becomes a bona fide abelian group for G abelian, whence we supress the ˆ, and the associated bundle, still presently principal, becomes a Z2 cohomology class. Now we have the induced exact cohomology sequence, i.e.
0 1 + 1 1 H (M,Z2) → Hˆ (M,GL ) → Hˆ (M,GL) → H (M,Z2) (3)
τ lives in the third group, and by exactness it has antecedent (i.e., M is orientable) if it goes to zero in the final group: the middle bundle above reduces if and only if the quotient splits: so we have the result:
M is orientable if and only if the first Stiefel-Whitney class of 1 τ, that is w1 ∈ H (M,Z2), is zero.
In other words: M is orientable if τ reduces its group GL to the connected subgroup GL+. According to the fundamental exactness relation, orientabil- ity means section in the lower bundle, and as it is principal, the bundle is trivial, hence its class (w1 ) is the zero of the cohomology group: the lower bundle in (2) splits.
2 Alternatively, M is orientable if it has a volume form, which is the same as a global frame mod det > 0 transformations in overlapping patches. As examples, RP 2 is not orientable, but RP 3 is orientable, where RP n is the real n-dimensional projective space of rays in Rn+1. The reason is the n n n antipodal map (−1, ..., −1) in S leading to RP = S /Z2 is a rotation for n odd, but a reflection for n even; note RP n is not simply connected for any n. To have 1-cohomology in any ring the first cohomology group H1(M,Z) has to be =6 0: simply connected spaces are orientable. Notice the structure group GL reduces always to the orthogonal group O = O(n): any manifold is, in its definition, paracompact, and in any para- compact space there are partitions of unity, hence for a manifold a Riemann metric is always possible:
O(n) ↓ GL(n, R) → B → M (4) ↓ Rn(n+1)/2 → E → M
As the lower row fibre is contractible, the horizontal middle bundle lifts; O(n) → B0 → M, where B0 is the set of orthogonal frames: any manifold is riemanizable. Note O(n) is the maximal compact subgroup of GL(n), and this is why the quotient is contractible. By contrast, not every manifold admits a Lorentzian metric: it needs to have a field of time-like vectors globally defined. 1 Hence the characteristic class w1 ∈ H (M,Z2) is the obstruction to ori- entability: it measures if an orientation is possible in a manifold. The next question is: If the obstruction is zero, how many orientations are there? Again, the answer is written by (3): the elements in Hˆ 1(M,GL+) falling 0 into τ are the coset labelled by H (M,Z2). In particular, if the manifold 0 is connected, H (M, A) = A, and then the zeroth Betti number is b0 = 1: 0 hence, as then H (M,Z2) = 2,
A connected orientable manifold has exactly two orientations.
The orthogonal group O(n) is neither connected nor simply connected; 0- connectivity questions lead to the first Stiefel-Whitney class, 1-connectivity
3 to the second (sw) class. Suppose the manifold M is orientable already and write the covering group Spin(n) → SO(n):
Z2 → Spin(n) → SO(n). (5)
For n> 2 this bundle is the universal covering bundle, as π1(SO(n)) = Z2, n> 2. For n = 2 the spin bundle still covers twice, but now π1(SO(2)) = Z. Endow now M with a riemannian structure (there is no restriction on doing this, see above), and write
Z2 = Z2 ↓ ↓ Spin(n) → B˜ → M (6) ↓ ↓ k τ : SO(n) → B → M
We say that a manifold admits an spin structure (it is spinable) if the (rotation) tangent bundle lifts to a spin bundle. Again, there is a precise homological answer. From the exact sequence, and with τ living in third group
1 1 1 2 H (M,Z2) → Hˆ (M, Spin(n)) → Hˆ (M,SO(n)) → H (M,Z2), (7) we see, as before, that if we call w2 the image of τ, there is an obstruction to spinability, called the second Stiefel-Whitney class,
2 w2 = w2(τ) ∈ H (M,Z2) (8) and a manifold is spinable if and only if w2(M) = 0. For example, spheres and genus-g surfaces are spinable. If the obstruction is zero, how many spin structures there are? Again, a simple look at (7) gives the answer: there 1 are as many as H (M,Z2), which is a finite set, of course. For example, for 1 2g 2g an oriented surface of genus g, H (Σg,Z2) = Z2 , hence # = 2 , as is well known in string theory; recall that the first Betti number b1(Σg)=2g. As examples of spin manifolds, CP 2n+1 is spinable, but CP 2n is not. For 1 2 2 example, CP = S , no sw classes; as for CP , we have that b2 = 1; recall also Euler number (CP n)= n + 1. There is an alternative characterization of spin structures due to Milnor [2], which avoids using Cech˘ cohomology sets. From (6) we have the exact sequence (taking values in Z2):
1 1 1 2 0 → H (M,Z2) → H (B,Z2) → H (SO(n),Z2) → H (M,Z2), (9)
4 and accepting w2 = 0 we see again the number of spin structures to be 1 #H (M,Z2), as B˜ is the total space of the lifted bundle, and lives in the second group.
3 The first Chern class
The Det map O(n) → O(n)/SO(n)= Z2 can be performed also in complex bundles, with structure group U instead of O: When does a complex bundle η reduce to the unimodular group? Write
SU(n) ↓ η : U(n) → B → M (10) ↓ ↓ k ′ c1 : U(1) → B → M 1 The associated bundle Det η ∈ H (M, U(1)) determines the first Chern class of η by the resolution Z → R → U(1) = S1, as
2 1 c1(η) ∈ H (M,Z)= H (M, U(1)). (11) c1(η) = 0 is the condition for reduction to the SU subgroup, an important restriction in compactifying spaces in M-theory, see later. For the general definition of Stiefel-Whitney (and Pontriagin) classes of real vector bundles, and for the Chern classes of complex vector bundles, the insuperable source is [3]
4 Euler class as Obstruction
A more sophisticated example is provided by the Euler class. Look for man- ifolds M with a global 1-frame, i.e. a global zeroless vector field: let M be orientable; from the coset Sn−1 = SO(n)/SO(n − 1)
SO(n − 1) ↓ τ : SO(n) → B → M (12) ↓ ↓k τ ′ : Sn−1 → B′′ → M A 1-frame exists if the (unit) sphere bundle has a section. The last bundle produces a map Hn−1(Sn−1, R) → Hn(M, R) (13)
5 The image of the fundamental class of Hn−1(Sn−1,Z) is the Euler class e of M. The condition of reduction is clearly that e = 0. Here e[M] = χ, the Euler number. The result is the well-known condition for a manifold to admit a global 1-frame: zero Euler number. It is funny (and easy to understand) that the theorem has a positive side: you can compute the Euler number by counting theWindungzahl of the zeros of any vector field, the Poincare-Hopf theorem.
5 Nonabelian Group Extensions
(Cfr. ([4]), Ch. 7). Consider the group extension problem: given groups K and Q, find G such that K ⊂ G normal and G/K = Q. Recall first the relations, for ZH : center of H
H/ZH ≡ Int H and Aut H/Int H ≡ Out H (14) for any group H. If there is a solution G/K = Q to our problem, write
ZK ↓ K → G → Q (15) ↓ ↓ ↓ IntK → AutK → OutK where you construct the last two vertical arrows. So any extension determines a morphism α : Q → OutK. Inverse question is: given α ∈ Hom(Q, OutK), are there extensions? How many? Note first, given α there is an extension
α : Int K → X → Q k ↓ α ↓ (16) Int K → Aut K → Out K That is
ZK ↓ K (17) ↓ Int K → X → Q We need to lift the horizontal sequence to have extensions, and we see that there is an obstruction in the exact cohomology sequence,
∗ ∗ ∗ ∗+1 H (Q, ZK) → H (Q, K) → H (Q, IntK) → H (Q, ZK) (18)
6 So α produces extensions iff the image of α in the last group is zero: this is the obstruction. Now if we add our knowledge that the abelian extensions 2 3 are given by some H , the obstructions lies in H (Q, ZK), and if it is zero, 2 the number of extensions is H (Q, ZK); and the obstruction lies in the third group: 3 w(α) ∈ H (Q, ZK); (19) all this is very similar to the spin or orientation problems.
6 Structure of Lie groups
An unexpected problem where an obstruction is necessary appears in the existence of simple Lie groups. Consider the next simplest group, SU(3). In the natural 3 representation, the group leaves the unit sphere invariant, with SU(2) as little group:
SU(2) = S3 → SU(3) → S5 ⊂ R6 = C3 (20) and regard this as a bundle extension; bundles over n-spheres are classified by πn−1(G), where G is the structure group. So here, as
3 π4(S )= Z2 (21) we have just two solutions, the direct product (which cannot be the group SU(3), because S5 is not paralellizable), and the other, necessarily SU(3): the existence of non-trivial budles, here (in this case) for nontrivial homotopy classes are crucial for the existence of Lie groups. Incidentally, the map S4 → S3 generating SU(3) is easy to describe: it is the suspension of the second Hopf bundle. (I thank D. Freed for this remark):
β : S1 → S3 → S2 Σ ↓ Σ ↓ (22) S4 → S3 For all Lie groups besides the “atom in the category”, SU(2) = S3 the same obstructions obtain; we leave the details. It would be nice to invert the question: to deduce the simple Lie groups from nontrivial extensions... Incidentally, the Hopf β bundle is the second on the series of higher homotopy 2n groups of spheres π4n−1(S )= Z+..., related to the Hopf invariant and to the nonexistence of division algebras besides R, C, H and O [5]. For expressions of simple compact Lie groups as finite twisted products of odd spheres, see [6]
7 7 Special Holonomy manifolds
Since the advent of M-Theory (1995; Townsend, Witten, Polchinski [7]) the problem of compactification of extra dimensions, from 10 to 4 in one extreme to 12 down to 2 in the other, is becoming more and more acute. If one is a true believer in M (or F ) theory (as I tend to be), this problem is perhaps the central one in physics. The arguments for extra dimensions are overwhelming, and so are the reasons why we live in four large dimensions. In a nutshell, geometric description of nongravitational forces requires extra dimensions, while interactions transmitted via massless particles do not make physical sense outside four (i.e. the 1/r potential law). Here we want to show, via simple examples, that compactification with extra conditions (like preserving N = 1 Supersymmetry) can be easily stated in cohomological terms, as reductions of the structure/holonomy group of different bundles. Consider the “old” ‘problem of compactifying the Heterotic String living in 10D down to 4D . The tangent bundle of the compactifying manifold K6 is τ : O(6) → B → M = K6 (23)
Now we want K6 to be a manifold orientable ( to integrate), spin (to describe fermions) and with a (covariant) constant spinor field (to preserve N = 1 Susy in order to “understand” the scale of the Higgs mass together with the existence of chiral fermions). In terms of reduction: O(6) reduces to SO(6); SO(6) lifts to Spin(6) = SU(4). SU(4) reduces to SU(3), which lies in U(3):
SU(3) ↓ τ˜ : U(3) → B → M (24) det ↓ ↓ k U(1) → B′ → M 1 Now R/Z = S induces (see above) det (ˆτ = c1(τ(M)), the first Chern class; hence M is a complex manifold with SU(3) holonomy, with the first Chern class =0, and it can be seen that this implies the trace of the curvature zero; it is a Ricci-flat riemannian manifold: Calabi-Yau manifolds. The search for those manifolds was a prolific industry led by Phil Candelas in Austin in 1985-92 [8]. Manifolds with tangent structure groups less than maximal are therefore crucial for M-theory; let us see more examples.
8 8 Compactification in M-theory
Please notice, first, that the reduced holonomy problem is not the same as reducing the structure group, but in practice both are present together; the link is of course the two holonomy theorems: [9] (1) The structure group can be reduced to the holonomy group, the Ambrose-Singer theorem (2) The Lie algebra of the holonomy group is generated by the curvature of the connection producing the holonomy in the first place, the curvature theorem. For a generic riemannian manifold, the possibility of isometry groups and reduced holonomy groups are antagonic: a generic manifold M has no isome- tries, and maximal holonomy (e.g. SO(dim M) if M orientable). Viceversa, special holonomy manifolds have no isometries in general (what poses a prob- lem for the existence of gauge groups down in 4D by the KK mechanism, see later), and a very symmetric space, in fact maximally symmetric, like even-dim spheres, has irreducible holonomy SO(n). M. Berger (1955) clasified holonomy groups, and came up with several series (like O(2n) ⊃ U(n),O(4n) ⊃ Sp(n) etc.), and just two (in fact, three; one, corresponding to Spin(9) ⊂ SO(16) was already known as a symmetric space) special cases:
Spin(7) ⊂ SO(8) and G2 ⊂ SO(7) (25)
Both turned out essential in M and F theories. Both come, of course, from the beautiful irreducible representations provided by the Clifford alge- bra. Notice this is irreducible holonomy, in the sense that the irrep of the subgroup has the same dim as that of the group, namely 8 and 7 bzw. To demystify those cases it is enough to ask for those representations of the spin groups Spin(n) which act trans in the unit sphere; and the answer is, besides the low dimensional cases in which there are repetitions (like Spin(6) = SU(4)), only two more: the 16 irrep of Spin(9) and the 8 of 2 Spin(7). The first case gives only the Moufang plane OP = F4/Spin(9). The other is very interesting: The 8 irrep of Spin(7) allows for the embeding Spin(7) ⊂ SO(8). Now SO(8) preserves a quadratic form. What else does Spin(7) mantain? What manifolds have Spin(7) holonomy? The construction of these manifolds start- ing by D. Joyce around 1995 [10] has been a great achievement. We shall comment on these constructions later, but let us finish first with the G2 case. In general, in a (real or complex) vector space V , the set X of 2-forms and/or quadratic forms are “open” in the sense of the orbit space X/GL(V )
9 ([11]). But p-forms dimension grows, of course, like n!/p!(n − p)! > n2 = dim GL(V ). The exceptions occur naturally in low dimensions: A three- form in 6, 7 and 8 dimensions and self-dual 4-form in 8 dimensions. In particular, the group leaving a regular 3-form invariant in 7-dim space is G2. (Note how the dimensions match: 49 = 7 × 7 = 35 (dim 3-forms) +14 (dim G2)). Which 3-form? Bilinear forms produce scalar products, but trilinear forms produce an internal law X × X → X, so one guesses that in R8 there is a product! Of course, there is: octonion multiplication. The apearance of octonions just reinforces the idea that M-theory, as a unique theory, has to include octonions, which are unique structures in mathemat- ics, and responsible for most of their exceptional objects [12]. To repeat: the reason of the appearance of octonions in M-theory is this: in the 11 to 4 version, the compact manifold has to have G2 holonomy becuse of su- persymmetry;this group preserves a 3-form, which corresponds to the fully antisymmetric (alternative) octonion multiplication 3-form. No wonder, G2 is the automorphim group of octonions. In the F -theory version, the 12 → 4 descent implies Spin(7) holonomy; 7 but this group can be seen as unit-octonion “group” S stabilized by G2. In both cases of 11 = (1, 10) dimensions and 12 = (2, 10) it is remarkable that supersymmetry unveils octonionion structures!
9 Structure Diagrams
It is time to express in diagrams what we are saying. First, there is the structure diagram for G2: define it as the little (=isotropy) group for the trans action of Spin(7) in the 7-sphere [5] :
SU(3) → SU(4) = Spin(6) → S7 ↓ ↓ k 7 G2 → Spin(7) → S (26) ↓ ↓ S6 = S6
So G2 operates in the 6-sphere of unit imaginary octonions; but G2 is also a subgroup of SO(7), witness the 7 irrep: the torsion diagram explains this:
Z2 = Z2 ↓ ↓ 7 G2 → Spin(7) → S (27) k ↓ ↓ 7 G2 → SO(7) → RP
10 and the mixed diagram for G2 SU(2) = SU(2) ↓ ↓ 6 SU(3) → G2 → S (28) ↓ ↓ k 5 6 S → V11 → S where V11 is a Stiefel manifold, generating the 2-torsion in G2, in the odd sphere structure [13]
3 11 G2 = S (×S (29) Finally, we exhibit the richness of the S7 sphere of unit octonions in the following special holonomy diagram
Sp(1) ⊂ SU(3) ⊂ G2 ⊂ SO(7) ↓↓↓ ↓ Spin(5) ⊂ Spin(6) ⊂ Spin(7) ⊂ SO(8) = Spin(8)/Z 2 (30) ↓↓↓ ↓ S7 = S7 = S7 = S7 Sp(H) U(C) Oct(O) O(R)
10 Structure of Special Holonomy manifolds
We return to a question mentioned above. In the old Kaluza-Klein approach, where MD → M4, one gets gauge forces in the lower space from the isometries of the compactifying manifold; for example, the U(1) for the electromagnetic field in the original 5 → 4 reduction of Kaluza (1919). But now, where extra dimensions are there to stay, the argument does not work anymore! Special holonomy manifolds have, generically, no isometries; so if we are to rely on simple gravity in higher spaces, how do we get gauge forces in our mundane 4D space? The answer is spectacular, and I do not think it has been assimilated wholly by the scientific community: the special holonomy manifolds have a rich homology, and the non-trivial cycles (that is, the uncontractible spheres) can act as sources for gauge fields, following the pattern of singularities, A − D − E classification, and Mac-Kay correspondence! [14], [15]. In a way, this is the generalization of the fact that the open string sustains gauge groups in its boundary, the singular points. Or, that compactification of M-theory in a segment necessitates two E8 groups in the border, the Horawa-Witten˘ mechanism.
11 We do not want to pursue a line of research which seems to be incomplete as yet. However, we cannot resist to consider the case of the perhaps simplest special (special but not exceptional) holonomy, the case of K3 (see an early example in [16]), which appears when the 11 → 7 or 10 → 6 descents; it is representative of the many sophisticated constructions of Joyce and others. So let us construct K3 [17] 1) Start with R4, divide by a lattice L to generate a 4-Torus
T 4 = R4/L = R4/(Z + Z + Z + Z) (31) 2) Now apply a discrete group Γ with a non-free action, for example the “parity” operation generating Z2:
θi → −θi (32) for the four angles labelling T 4. Call X = T 4/Γ. The space X is an orbifold, that is, a manifold with some special (singular) points, those fixed by Γ (here there are 24 = 16 points). 3) The 4-Torus is a complex surface, the X space is also a complex surface, and as a complex manifold, there is a perfectly standard procedure (starting in the 19th century by italian mathematicians!) to remove (blow-up) the singularities, trading them, in our case, by 2-spheres (complex projective lines, really). The resulting true, bona fide smooth manifold is called K3 in the literature (for Kummer, K¨ahler and Kodaira, amen for coincidence with the Himalaya peaks; the godfather seems to be A. Weil [17]). The reader can think of converting the cone x2 + y2 = z2 on the one- sheeted hyperboloid x2 + y2 − z2 = 1 as a simple blow-up of the conic singu- larity, trading it by a circle. It is useful to pursue the change in topology: the Betti numbers are (1, 0, 0, 0, 0) for R4, (1, 4, 6, 4, 1) for T 4, (1, 0, 6, 0, 1) for X and (1, 0, 22, 0, 1) for K3. Notice the change after the blow-up: each of the 16 singular points fattens to become a “hollow” 2-sphere, hence b2 increases from 6 to 22; notice also the increase in ”curvature” from T 4, which is flat: already in X there is “point” curvature. From the point of view of string theory, the point of introducing K3 is that string theory IIA or IIB dualizes with the heterotic string in this curious way, in six dimensions [19]:
II/K3 ≈ Het/T 4 (33)
12 In other words, K3 “generates” whatever remains in 6D of the 496-dim gauge group extant in 10D! We do not enter into details, as they are well known, albeit not well understood.
11 Acknowledgements
This is a slightly worked out edition of the talk I gave in Salamanca. It is meant to recall the past happy times forty years ago when Adolfo and myself were together in Barcelona learning the rudiments of cohomology (mainly from Prof. Juan Sancho Guimera), with applications to projective representations and group extensions.
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